Method to Automatically Calibrate a Downhole Tool in an Oil-Based Mud Environment

ABSTRACT

A method and apparatus to calibrate a resistivity measurement taken by a downhole tool in a borehole, wherein the downhole tool estimates the resistivity of an underground formation penetrated by the borehole with at least one sensor situated at a non-zero standoff distance from the borehole, is provided. The method includes taking apparent impedance measurements with the sensor at a set of frequencies and at a first plurality of locations in the borehole, wherein the measurement are uncalibrated measurements. The method also includes identifying a portion of the borehole in which the apparent impedance measurements at least at a first frequency of the set have a predetermined behavior. The predetermined behavior is that the apparent impedance measurements taken in the portion are substantially fitting a linear model when represented in the complex plane. The method also includes using a plurality of measurements obtained at a second plurality of location situated in said portion at the first frequency to determine calibration coefficients for the measurements at said frequency.

BACKGROUND

The present disclosure relates to techniques for performing formationevaluation. More particularly, the present disclosure relates totechniques, such as calibrations, that may be used in performingmeasurement, imaging and/or other formation evaluations.

To locate and capture valuable hydrocarbons from subterraneanformations, various wellsite tools may be used to perform various tasks,such as drilling a wellbore, performing downhole testing and producingdownhole fluids. Downhole drilling tools may be advanced into the earthby a drill string with a bit at an end thereof to form the wellbore.Drilling muds (or other drilling fluids) may be pumped into the wellboreand through the drilling tool as it advances into the earth. Thedrilling muds may be used, for example, to remove cuttings, to cool thedrill bit and/or to provide a coating along the wellbore. The drillingmuds may be conductive or non-conductive drilling fluids (e.g., oilbased muds (OBM), water based muds (WBM), etc.) During or afterdrilling, casing may be cemented into place to line a portion of thewellbore, and production tools may be used to draw the downhole fluidsto the surface.

During wellsite activities, downhole measurements may be taken tocollect information about downhole conditions. The downhole measurementsmay be taken of various wellsite parameters, such as temperature,pressure, permittivity, impedance, resistivity, gain factor, buttonstandoff, etc. Downhole tools, such as the drilling tool, a testingtool, a production tool, or other tools, may be deployed into thewellbore to take the downhole measurements, such as formationresistivity. In some cases, downhole logs, images or other outputs maybe generated from the downhole measurements.

However, the downhole measurement that is taken generally does not onlycharacterize the formation: it is affected by the sensor itself but alsoby the drilling mud that is situated in the borehole, in particular whenthe tool is a LWD tool for which the distance between the sensor and theformation (ie standoff) is high. The tool performing the measurementstherefore needs to be calibrated in order to have the measurementcharacterize at best the formation and eliminate influence of the otherelements on the measurements.

Generally, the tools are calibrated at the workshop, and the calibrationdoes not take into account the effect of the drilling fluid that is inthe wellbore when the measurement is performed. However, when thedrilling fluid is a non-conductive fluid such as oil-based mud, it has avery significant impact on the measurement and the accuracy of themeasurement is greatly improved when the drilling fluid is taken intoaccount in the calibration.

US 2014/0347056 discloses calibrating a tool used for a downholemeasurement in-situ in the wellbore casing, before reaching an open holeportion of the borehole for a particular tool. This method is notappropriate for LWD tools.

SUMMARY

This summary is provided to introduce a selection of concepts that arefurther described below in the detailed description. This summary is notintended to identify indispensable features of the claimed subjectmatter, nor is it intended for use as an aid in limiting the scope ofthe claimed subject matter.

The disclosure relates to a method and apparatus for calibrating aresistivity measurement taken by a downhole tool in a borehole, whereinthe downhole tool estimates the resistivity of an underground formationpenetrated by the borehole with at least one sensor situated at anon-zero standoff distance from the borehole, is provided. The methodincludes taking apparent impedance measurements with the sensor at a setof frequencies and at a first plurality of locations in the borehole,wherein the measurement are uncalibrated measurements. The method alsoincludes identifying a portion of the borehole in which the apparentimpedance measurements at least at a first frequency of the set have apredetermined behavior. The predetermined behavior is that the apparentimpedance measurements taken in the portion are substantially fitting alinear model when represented in the complex plane. The method alsoincludes using a plurality of measurements obtained at a secondplurality of location situated in said portion at the first frequency todetermine calibration coefficients for the measurements at saidfrequency.

The calibration method according to the disclosure enables to calibrateLWD tools encountering multiple standoffs in any type of mud, evenoil-based mud, directly in the borehole, considering the characteristicsof the tool but also of the environment of the borehole (in particularthe nature of the drilling fluid). It enables to obtain an accuratemeasurement without performing extra separate calibration operation.

The disclosure also relates to an apparatus for calibrating aresistivity measurement, wherein the apparatus includes a downhole toolconfigured to be conveyed in a borehole, and having at least one sensorsituated at a non-zero standoff distance from the borehole. The sensoris configured to estimate the resistivity of an underground formationpenetrated by the borehole by taking apparent impedance measurements ata set of frequencies and at a first plurality of locations in theborehole, wherein the measurement are uncalibrated measurements. Theapparatus also includes a set of processors configured to identify aportion of the borehole in which the apparent impedance measurements atleast at a first frequency of the set have a predetermined behavior. Thepredetermined behavior is that the apparent impedance measurements atthe first frequency taken in the portion are substantially fitting alinear model when represented in the complex plane. The set ofprocessors is also configured to use a plurality of measurementsobtained at a second plurality of location situated in the portion atthe first frequency to determine calibration coefficients for themeasurements taken at the first frequency.

BRIEF DESCRIPTION OF THE DRAWINGS

Various aspects of this disclosure may be better understood upon readingthe following detailed description and upon reference to the drawings inwhich:

FIG. 1 is a schematic drawing of a drilling system including theapparatus as per the disclosure,

FIGS. 2 and 3 are views of a sensor of the apparatus for measuringresistivity of an underground formation according to an embodiment ofthe disclosure,

FIG. 4 is a model of electrical circuit used for modelling measurementsvia the apparatus according to the disclosure in the formation,

FIG. 5A is a flowchart showing an embodiment of the method according tothe disclosure,

FIG. 5B-5D are flowcharts showing details of operations of the method ofFIG. 5A,

FIG. 6 is a model of electrical circuit used for modelling measurementsvia the apparatus according to the disclosure in the casing,

FIG. 7 is a representation of an impedance measurement in a complexplane,

FIG. 8 shows representation of measurements taken with the apparatusaccording to the disclosure in the complex plane,

FIG. 9 is a flowchart showing another embodiment of the method accordingto the disclosure,

FIG. 10 is a flowchart showing another embodiment of the methodaccording to the disclosure

DETAILED DESCRIPTION

One or more specific embodiments of the present disclosure will bedescribed below. These described embodiments are examples of thepresently disclosed techniques. Additionally, in an effort to provide aconcise description of these embodiments, some features of an actualimplementation may not be described in the specification. It should beappreciated that in the development of any such actual implementation,as in any engineering or design project, numerousimplementation-specific decisions may be made to achieve the developers'specific goals, such as compliance with system-related andbusiness-related constraints, which may vary from one implementation toanother. Moreover, it should be appreciated that such a developmenteffort might be complex and time consuming, but would still be a routineundertaking of design, fabrication, and manufacture for those ofordinary skill having the benefit of this disclosure.

When introducing elements of various embodiments of the presentdisclosure, the articles “a,” “an,” and “the” are intended to mean thatthere are one or more of the elements. The terms “comprising,”“including,” and “having” are intended to be inclusive and mean thatthere may be additional elements other than the listed elements.Additionally, it should be understood that references to “oneembodiment” or “an embodiment” of the present disclosure are notintended to be interpreted as excluding the existence of additionalembodiments that also incorporate the recited features.′

FIG. 1 is a schematic diagram of a drilling system 10, which may be usedto drill a well or borehole through a geological formation 12. In thedepicted example, a drilling rig 14 at the surface 16 rotates a drillstring 18, which includes a drill bit 20 at its lower end to engage thesub-surface formation 12. To cool and/or lubricate the drill bit 20, adrilling fluid pump 22 may pump drilling fluid, referred to as “mud” or“drilling mud,” downward through the center of the drill string 18 inthe direction of the arrow 24 to the drill bit 20. At the drill bit 20,the drilling fluid may then exit the drill string 18 through ports. Thedrilling fluid may then flow in the direction of the arrows 28 throughan annulus 30 between the drill string 18 and the geological formation12 toward the surface 16. In this manner, the drilling fluid may carrydrill cuttings away from the bottom of a borehole 26. Drill cuttings or“cuttings” include small pieces of rock or other debris that break awayfrom the geological formation 12 as a result of drilling. Once at thesurface 16, the returned drilling fluid may be filtered and conveyedback to a mud pit 32 for reuse.

Additionally, as depicted, the lower end of the drill string 18 includesa bottom-hole assembly 34 that includes the drill bit 20 along with adownhole tool 36, such as a measuring tool, a logging tool, or anycombination thereof. Generally, the downhole tool 36 may facilitatedetermining characteristics of the surrounding formation 12. Thus, insome embodiments, downhole tool 36 may include one or more sensors 42.Further references to the sensor 42 may refer to one or more sensors 42of the downhole tool 36. In some embodiments, the sensor 42 may includean acoustic sensor (for instance, an ultrasonic pulse-echo transducer),which may perform acoustic measurements returned from the surroundingformation 12. In some embodiments, the sensor 42 may include anelectrical sensor (for instance, an electromagnetic transducer orreceiver), which may perform electrical measurements (such as galvanicor inductive electrical measurement) returned from the surroundingformation 12.

As shown on FIG. 1, the borehole may be cased in its top portion 52, iea casing 54 has been added to surround the borehole and is attached toformation with cement (not shown) situated between the casing and theformation. The casing insulates the borehole from the formation andconsolidates the borehole. In this portion of the borehole, the drillingfluid does not contact the formation. The casing is generally a metallictubing. On the contrary, in its bottom portion 58, the borehole is openhole, ie the drilling fluid circulating in the borehole directlycontacts the formation. The measurements enabling to characterize theformation are generally taken in the open hole portion 58 of theborehole.

In some embodiments, a control system 44 may control operation of thedownhole tool 36. For example, the control system 44 may instruct thedownhole tool 36 to perform measurements using the sensor 42 and/orprocess the measurements to determine characteristics of the surroundingenvironment (e.g., formation 12). In some embodiments, the controlsystem 44 may be included in the downhole tool 38. In other embodiments,the control system 44 may be separate from the downhole tool 36, forexample, situated in another downhole tool or at the surface 16. Inother embodiments, a portion of the control system 44 may be included inthe downhole tool 36 and another portion may be located separate fromthe downhole tool 36.

When at least a portion is separate from the downhole tool 36,information (e.g., measurements and/or determined characteristics) maybe transmitted to and/or within the control system 44 for furtherprocessing, for example, via mud pulse telemetry system (not shown)and/or a wireless communication system (not shown). Accordingly, in someembodiments, the downhole tool 36 and/or the control system 44 mayinclude wireless transceivers 50 to facilitate communicatinginformation.

To facilitate controlling operation, the control system 44 may includeone or more processors 46 and one or more memory devices 48. Furtherreferences to “the processor 46” are intended to include the one or moreprocessors 46. In some embodiments, the processor 46 may include one ormore microprocessors, one or more application specific processors(ASICs), one or more field programmable logic arrays (FPGAs), or anycombination thereof. Additionally, the memory 48 may be a tangible,non-transitory, machine-readable medium that stores instructionsexecutable by and data to be processed by the processor 46. Thus, insome embodiments, the memory 48 may include random access memory (RAM),read only memory (ROM), rewritable flash memory, hard drives, opticaldiscs, and the like.

The downhole tool may comprise a sensor 42 shown in FIGS. 2 and 3 usedfor measuring resistivity of the formation 12. The sensor disclosed inthe figure is exemplary. The sensor 42 comprises or is otherwise carriedwith a tool collar 205. The tool collar 205 generally comprises atubular member having interfaces (not shown) at one or both ends forcoupling with other components of a tool string. The sensor 42 comprisesa probe 210 having an exterior surface 211 that may be substantiallyflush with an exterior surface 206 of the tool collar 205. For example,the probe 210 may be received within a recess or other opening 207 inthe exterior surface 206 of the tool collar 205. The probe 210 may beextendable away from the tool collar 205, whether via known orfuture-developed means, for instance a wireline pad.

The probe 210 comprises a button electrode 220, an inner or first guardelectrode 230 surrounding the button electrode 220, and an outer orsecond guard electrode 240 surrounding the inner guard electrode 230.Insulating material 250 electrically isolates the button electrode 220,the inner guard electrode 230, and the outer guard electrode 240 fromeach other and from a body 212 of the probe 210. The probe 210 alsocomprises one or more return electrodes 260, which are each alsoisolated from the body 212 of the probe 210 by insulating material 250.However, the return electrodes 160 may be formed by at least portions ofthe tool collar 205 instead of as discrete members carried by the probe210.

FIG. 4 also schematically depicts electrical components and connectionsbetween the elements described above. For example, one side of anexcitation voltage source 270 is connected to one of the returnelectrodes 260 and local circuit ground, with the other side of theexcitation voltage source 270 connected to the outer guard electrode240. The other one or more return electrodes 260 are also connected tolocal circuit ground. A sampling resistor 280 having resistance RBOG(first impedance) connects the button electrode 220 to the outer guardelectrode 240, and an additional resistor 285 having resistance RIGOG(second impedance) connects the inner guard electrode 230 to the outerguard electrode 240. The sampling resistor 280 and the additionalresistor 285 may be positioned in a housing 296, such as may bedelimited by the outer guard electrode 240.

FIG. 3 also illustrates an acquisition board 297 disposed within thetool 200. The housing 296 may contain the acquisition board 297, may becoupled with the acquisition board 297, or may be a distinct componentseparate from the acquisition board 297 but having one or moreelectronic components coupled with the acquisition board 297, such asthe sampling resistor 280 and/or the additional resistor 185.

The sensor 42 shown in FIGS. 2-3 may be used for measuring resistivityof the formation 12. During such operations, an alternating current isapplied between outer guard electrode 240 and a return electrode 260 viathe voltage source 270. The voltage may be a high-frequency voltage,such as a frequency higher than about 100 kHz, or perhaps higher thanone MHz, or even ten MHz. Then, the current circulating through thesampling resistor 280 is measured. On the basis of the measured current,an measured impedance Z_(f) may be determined and related to theimpedance Z_(form) of the formation 20.

The sensor 42 disclosed here is a sensor according to an embodiment ofthe disclosure. A sensor according to such embodiment is disclosed inmore details in WO2016/082925, hereby incorporated by reference in thecurrent application. The tool according to the disclosure may alsoinclude other type of resistivity sensors. It is also understood thatthe downhole tool may also include a plurality of resistivity sensors42, wherein the sensors are identical or different.

When the measurement with sensor 42 is taken is the open hole portion ofthe borehole, it is considered that the response of the resistivity toolcan be modelled by a complex equivalent circuit as illustrated in FIG.4. The complex equivalent circuit is formed as a first mud impedanceZ_(mud1) coupled in series with a measured impedance Z_(f)representative of the formation impedance Z_(form) and coupled inparallel with a second mud impedance Z_(mud2), as illustrated by theequivalent circuit shown in FIG. 4. The formation impedance Z_(form) isdetermined based on the measured total impedance Z_(app).

The circuit model illustrated in FIG. 4 comprises the impedance Z_(f),representative of the formation impedance Z_(form), and a first mudimpedance Z_(mud1) in series, and also a parallel second mud impedanceZ_(mud2). The enhanced circuit model effectively represents a currentleakage that occurs as the standoff distance between electrodes of theresistivity tool and the subterranean formation increases—which is thecase when the measurement is a LWD measurement taken while drilling.Some of the current paths measured by the resistivity tool are indeednot passing through the subterranean formation. They correspond to theelectrical branch comprising the second mud impedance Z_(mud2). Othercurrents are passing through the formation and the drilling fluid, whichcorrespond to the electrical branches comprising the measured impedanceZ_(f) and the first mud impedance Z_(mud1). This circuit model thustakes into account the standoff distance between the electrodes of theresistivity tool and the surrounding subterranean formation, whichdistance may not be accurately known and/or which may corrupt aresistivity estimate of the subterranean formation.

In order to calculate the formation impedance Z_(form) in view of themeasured impedance Z_(f) the measurement is taken at a plurality offrequencies that are preferably spanning a large range of frequencies,such as for instance [0.2, 400] megahertz (“MHz”). The method fordetermining the resistivity of the formation with a resistivity toolusing such model is disclosed in more details in US2017/0227666, herebyincorporated by reference. However, the method for determining theresistivity described herein is an exemplary method. The calibrationmethod according to the disclosure may be applicable to any tool usingan electrical model similar to the one disclosed above for determiningresistivity of the formation.

As explained hereinabove, such measurement needs to be calibrated toeliminate the effect of the sensor on the measurement. Furthermore, thetool must be calibrated to be able to characterize the formation nomatter what the type of drilling fluid (ie mud) circulating in theformation is. An in-situ calibration in the casing enables to increasethe measurement accuracy as the tool may be calibrated in the same mudthat will be circulating in the borehole while the measurement of theformation resistivity will be performed. As the method that is used andincludes a plurality of frequencies spanning a broad range, thecalibration shall as well take the mud dispersion, ie the frequencyresponse of the mud impedance, into account. The mud impedance as afunction of the frequency is expressed as follows:

$\begin{matrix}{Z_{mud} = {\alpha \; X_{m}\frac{F(\omega)}{\omega}}} & (1)\end{matrix}$

wherein Z_(mud) is the mud impedance, w is the radial frequency, αX_(m)is a complex number depending on the standoff and F(ω) is afrequency-dependent dispersion function. The calibration methoddiscloses therein enables to calibrate automatically the measurements atall of the frequencies, taking into account the sensor and mud effectson the measurements.

The calibration method 300 according to the disclosure will be disclosedin reference to FIG. 5A. It comprises taking (block 302) a plurality ofmeasurements in the borehole at a plurality of locations while the toolis lowered in the borehole. As indicated, the borehole has a top casedhole section 52 and a bottom open hole section 58 in which the drillingfluid directly contacts the formation. The circuit model modelling thetool response in the formation has been already explained. In the casedhole portion of the borehole however, the tool does not sense theformation but rather sense the casing that is very conductive due to itsmetallic composition. Therefore, the equivalent circuit comprises afirst mud impedance Z_(mud1), and also a parallel second mud impedanceZ_(mud2) as represented on the model of FIG. 6. Some of the currentpaths measured by the resistivity tool are indeed not passing throughthe casing. They correspond to the electrical branch comprising thesecond mud impedance Z_(mud2). Other currents are passing through thecasing and the drilling fluid, which correspond to the electricalbranches comprising the first mud impedance Z_(mud1). The casing isconsidered as conductive enough to have no impedance contrary to theformation.

The calibration method then comprises representing all of the measuredimpedance in the complex plan (block 304). Indeed, in view of theabove-mentioned models, when the tool is in the cased hole portion ofthe borehole, as Z_(mud1) and Z_(mud2) are proportional to the mudimpedance Z_(mud), the measured impedance linearly depends on the mudimpedance. The measured impedance also depends on the standoff. On thecontrary, when the measurements are taken in the formation, they do notanymore linearly depend on the mud impedance. Such representation in thecomplex plane is shown on FIG. 7. The representation 350 in the complexplan shows the measurement point Z 352 in the complex plan. The realpart Re(Z) of the measured impedance Z is shown on the ordinate axis 354and the imaginary part Im(Z) of the impedance in abscissa 356. Thecomplex representation also enables to determine the module and phase ofthe impedance by tracing a line 358 between the measured impedance andthe origin of the axis, the module |Z| being the distance between themeasurement point and the origin while the phase is the angle θ betweenordinate axis and line.

When representing all of the measured impedance in the complex plane,the calibrated measurements taken in the casing (taken first while thetool is lowered in the wellbore) will be approximately on a same line,with a same phase for each of the measured frequencies. The module ofthe measured impedance will also be dependent on the standoff. When themeasurements are taken at several frequencies, the measurements at afirst frequency may be represented separately from the measurements at asecond frequency. In view of the above, when the measurements are takenat a plurality of frequencies, lines representing measurement atdifferent frequencies may have the same slope.

The calibration method may then include building a statistical linearmodel representative of the measurement in casing based on a first setof measurements corresponding to the first measurements acquired in theborehole, for each of the frequency (block 306). This may be performedvia a classic linear regression. A representation of a plurality ofmeasurements 360 taken in casing at one frequency are represented onFIG. 8. It can be seen that the measurements perfectly fit the line 362.

The calibration method may then include determining if each newmeasurement is taken in the casing (block 308). When a new measurementis acquired, a criterion is assessed in order to determine if it can betaken into account to refine to linear model. The criterion may be thatits distance from the resulting line is not greater than a threshold,or, taking into account several frequencies, that the difference betweenslopes of the lines representative of the measured resistivity at leastat two different frequencies is not greater than a predeterminedthreshold. Of course, a combination of several criteria may be assessed.The beginning of the open hole section may also be identified based onthe one or more criteria, i.e. when the one or more criteria are not metanymore. As it is well known that the cased hole and open hole portionsof the borehole are not intricated and that the open hole section alwaysfollows the cased hole section, the open hole section may be detectedvia determining that a predetermined number of consecutive measurementsdo not match the one or more criteria relative to the predeterminedlinear model, as defined above.

The calibration method then includes selecting a calibration set ofmeasurements (block 310) upon determining that the open hole section 58has begun. This is fairly simple as all the measurements before the openhole section may be taken into account as the calibration sample. Ofcourse, in order to have a more robust set of measurement points, anumber of measurement points just before the open hole section has beenreached may be discarded.

Before performing the calibration, the method may include validating thecalibration set (block 312). This operation includes verifying that theset of measurements is representative of all of the conditions that maybe found in a borehole, for instance a great diversity of standoffs. Asit can be shown that the module of the measured impedance is dependentof the standoff measurement, a measured impedance representative of aminimal standoff (ie having a low module) may be compared to a measuredimpedance representative of a maximal standoff (ie having a highmodule). On FIG. 8, the measured impedance that will serve as a basisfor the validation are represented in 364 and 366. They correspond forinstance to the measurements having a module in the n^(th) and(100−n)^(th) quantile (with n being preferably less or equal to 10). Theverification includes comparing one or more variable representative ofthe difference between both impedance (using a difference or a ratio forinstance) to one or more corresponding predetermined threshold. Eachthreshold may be a constant or may depend of one or more features of theline statistically representing the measurements in the complex plane.The comparison may include verifying that the standoffs are differentenough and/or if the measurements have a sufficiently wide distributionto be greater than the error fit. Based on the result, the calibrationset may be considered as valid for an accurate calibration or not. Ifthe calibration is not validated, default calibration coefficients maybe applied to the measurements (block 314) and other calibration set maybe looked for in the open hole section of the borehole (block 316), aswill be explained later in reference to FIG. 9.

Alternatively, the calibration method according to the disclosure maydetect that the conditions for the standoff diversity of a calibrationare met, define the calibration sample on this basis and verifyafterwards that all of the measurements (or only the measurement of theset taken last) is still in the casing to validate the calibrationsample.

Further, the method 300 is disclosed as performed in real-time. However,the method may be performed once the entire set of measurement for theborehole has been obtained, as post-processing, in which case themeasurements may not be evaluated one after another.

The method according to the disclosure therefore offers a calibration onthe basis of a great number of measurements that is robust and withoutany need to trigger the measurement of well-chosen calibration pointsfrom the surface. When validating the measurement with additionalcriteria such as the standoff diversity, it enables to make sure thatthe calibration will be representative of all of the conditions that maybe found in the borehole and therefore will enable accurate measurement.

Once the calibration sample has been validated, the method includesexpressing (block 318) the calibration parameters (also designated asthe calibration coefficients). The calibration coefficients may beexpressed as part of a linear model. Such coefficients will account formost of the effects due to the tool and the drilling fluid on themeasurement. However, other type of model might be used, for instancepolynomial models having a greater order than 1. As an example,calibration coefficients may be expressed as follows:

ZAPPa_CAL_Fb=g _(a,b)·ZAPPa_UNC_Fb+u _(a,b) , a=1 . . . N; b=1 . . .M  (2)

wherein ZAPPa_CAL_Fb is the calibrated measurement (unknown) for thesensor a and frequency b (for a downhole tool having N sensors operatingat M frequencies), ZAPPa_UNC_Fb is the uncalibrated measurement (known)for the sensor a and frequency b, and g_(a,b) and u_(a,b) are complexparameters that the calibration operation seeks to determine. It isimportant to note that g_(a,b) and u_(a,b) have different values foreach of the N sensors and each of the M frequencies.

The method then includes determining (block 320) the calibrationcoefficients ga,1 and ua,1 for measurements taken at a first referencefrequency f1, which is a low frequency, i.e. a frequency under athreshold (about 10 MHz for the sensor presented hereinabove but thethreshold value may depend on architecture of the sensor). Frequency f1is for instance the lowest measurement frequency. The details ofoperation 320 are represented on FIG. 5B. The method takes a firsthypothesis, which is the following: the line statistically representingthe measurements should, once calibrated, contain the origin of the axis(when there is no standoff, the tool is directly in contact with thecasing and does not measure any impedance). A point on the line istherefore chosen as the point that should be the origin of the axis.Here, the selected point is the point 368 being at minimal distance i.e.the orthogonal projection from the origin of the axis but other pointsmay be chosen. All the measurements of the sensor at frequencies 1 . . .M are calibrated as a function of the calibration hypothesis at thereference frequency f1. In other words, the method includes determininga first relationship between the calibration coefficients for themeasurement at the first reference frequency by correlating a pointtaken on the statistical linear model with the origin of the complexplan (block 322).

Determining the calibration coefficients ga,1 and ua,1 for measurementstaken at the first reference frequency also includes comparing (block324) theoretical impedance in a non-dispersive reference medium, such asair, to uncalibrated measurement in this medium obtained with the sensora and determining (block 326) a second relationship between thecalibration coefficients for the measurement at the first referencefrequency based on such comparison. The uncalibrated measurement mayhave been taken before the job once and for all and may be re-used ateach new calibration and/or may be modelled in view of the toolparameters. For low frequencies, it is indeed considered that themeasurement is not significantly affected by the mud dispersion orsensor size and that the calibration coefficients are the same orproportional for the reference medium and mud. Determining thecalibration coefficients for a low frequency then includes deriving(block 328) from the first and second relationship both calibrationcoefficients g_(a,1) and u_(a,1).

The determination 320 may be performed for all of the sensors that aresituated in the borehole and have to be calibrated. Therefore, theoutput of the determination 320 may be all of the coefficients g_(a,1)and u_(a,1) with a=1 . . . N.

The method may also include determining (block 330) the calibrationcoefficients g_(a,2) and u_(a,2) for measurements taken at least asecond frequency f2 lower than the predetermined threshold. Thedetermination 324 may also be performed for all of the frequencies fjunder the threshold.

The details of the operation 330 are represented on FIG. 5C. The methodfirst comprises operations 324 and 326 already disclosed above, whichwill give a first relationship between the calibration coefficientsg_(a,2) and u_(a,2).

However, as all of the measurements must be calibrated as a function ofthe calibration at the reference frequency f1 in order to have coherentmeasurements, it is not possible to use the same operation that has beenperformed at 322. It would indeed lead to a non-coherent calibration.Therefore the calibration coefficients g_(a,2) and u_(a,2) aredetermined using the calibration coefficients g_(a,1) and u_(a,1)obtained for the first reference frequency.

The method includes correlating (block 332) uncalibrated measurementstaken at the reference frequency f1 and uncalibrated measurement takenat second frequency f2. In view of the linear nature of the response, arelationship may be found between the uncalibrated measurement at bothfrequency, that is mathematically expressed as follows:

ZAPPa_UNCAL_F2=p _(a21)·ZAPPa_UNCAL_F1+q _(a21)  (3)

wherein ZAPPa_UNCAL_F2 Fb is the calibrated measurement (known) for thesensor a and frequency f2, ZAPPa_UNC_F1 is the uncalibrated measurement(known) for the sensor a and frequency f1, and p_(a21) and q_(a21) arecomplex parameters.

The method also includes correlating the calibrated measurements (block334) by expressing the impedance at the second frequency f2 as afunction of the impedance of the first reference frequency f1. As theimpedance measured at each frequency are both depending on the mudimpedance, for each frequency j, the impedance may be expressed asfollows:

${{{ZAPPa\_ CAL}{\_ Fj}} = {\alpha \; X_{m}\frac{F( {\omega \; j} )}{\omega \; j}}},$

as indicated above, wherein ωj is the radial frequency (ωj=2πfj), theimpedance at frequency f2 may be expressed as follows:

$\begin{matrix}{{{ZAPPa\_ CAL}{\_ F2}} = {{ZAPPa\_ CAL}{\_ F1} \times \frac{F( {\omega \; 2} )}{\omega \; 2} \times \frac{\omega \; 1}{F( {\omega \; 1} )}}} & (4)\end{matrix}$

This relationship includes one additional unknown parameter which isF(ω2). Indeed, F(ω1) is known from the previous computation fromequation

${{ZAPPa\_ CAL}{\_ F1}} = {\alpha \; X_{m}{\frac{F( {\omega \; 1} )}{\omega \; 1}.}}$

As frequency f1 is taken as the reference frequency, it is consideredthat |F(ω1)|=1 and the phase of F(ω1) is related to the phase ofZAPPa_CAL_F1 as the term αX_(m) only has an influence on the module ofthe impedance (as shown by the linear measurements). When thecalibration includes the calibration of several sensors, it may beinteresting to determine the dispersion function of the mud as afunction of the different sensors. Indeed, the properties of the mudshould be the same for all of the measurements. Therefore, the phase ofF(ω1) may be defined as the average of the phases obtained at firstfrequency f1 for all of the calibrated sensors. This gives morerobustness to the calibration.

The method then comprises determining a second and third relationshipbetween the calibration coefficients g_(a,2) and u_(a,2) and F(ω2) basedon the correlations of uncalibrated measurements performed at 332 and ofthe calibrated measurements performed at 334 (block 336). This operationis performed first by determining the complex parameters p_(a21) andq_(a21) which is made possible by using a plurality of uncalibratedmeasurements taken in the casing, and then to determine a relationshipbetween the coefficients g_(a,2), u_(a,2) and F(ω2) based on p_(a21) andq_(a21) and g_(a,1) and u_(a,1) using also the correlations (equations(3) and (4)) as well as the relationship for each frequency betweencalibrated and uncalibrated measurements (see equation (2) above).

Determining the calibration coefficients then includes deriving (block338) from the first, second and third relationships both calibrationcoefficients g_(a,2) and u_(a,2).

The determination 330 may be performed for all of the sensors that aresituated in the borehole and have to be calibrated. Therefore, theoutput of the determination 330 may be all of the coefficients g_(a,2)and u_(a,2) with a=1 . . . N.

The method may also include determining (block 340) the calibrationcoefficients g_(a,3) and u_(a,3) for measurements taken at least a thirdfrequency f3 higher than the predetermined threshold. The determination336 may also be performed for all of the frequencies fj above thethreshold. This determination cannot use the measurement innon-dispersive medium to calibrate the measurement at high frequency asother parameters of the tool or the environment may have a higherinfluence at such frequencies. The details of such operation are shownon FIG. 5D.

The determination 340 first comprises modelling (block 342) the muddispersion function. The mud dispersion function may be modelled usingany appropriate model.

The determination 340 then comprises determining (block 344) the unknownparameters of the mud model using the pre-determined values of F(ω1) andF(ω2) and mud dispersion value for any other frequency below thepredetermined frequency threshold defined above. As explained above forF(ω1), F(ω2) may be determined taking into account the values of thedispersion function obtained for several sensors. The method thereforeincludes obtaining (block 346) a value of the function F(ω3) for thefrequency f3 (it is reminded that ω3=2πf3). Of course, when thecoefficients are sought for several frequencies above the threshold,operation 342, 344 may be performed once and taken into account in thedetermination of the calibration coefficients for each frequency.

The determination also comprises correlating uncalibrated measurementstaken at one of the frequencies for which the calibration has alreadybeen performed (f1 or f2) and uncalibrated measurement taken atfrequency f3, as explained in relationship with operation 332. Based onthe measurement taken at frequency f1, the correlation may be expressedas follows:

ZAPPa_UNCAL_F3=p _(a31)·ZAPPa_UNCAL_F1+q _(a31)  (5)

wherein ZAPPa_UNCAL_F3 is the uncalibrated measurement (known) for thesensor a and frequency f3, ZAPPa_UNC_F1 is the uncalibrated measurement(known) for the sensor a and frequency f1, and p₃₁ and q₃₁ are complexparameters. It also comprises correlating calibrated measurements atfrequency f3 and calibrated measurement taken at one of the frequenciesfor which the calibration has already been performed (f1 or f2), asexplained in relationship with operation 334. Based on the measurementtaken at frequency f1, the correlation may be expressed as follows:

$\begin{matrix}{{{ZAPPa\_ CAL}{\_ F3}} = {{ZAPPa\_ CAL}{\_ F1} \times \frac{F({\omega 3})}{\omega \; 3} \times \frac{\omega \; 1}{F( {\omega \; 1} )}}} & (6)\end{matrix}$

In view of the model previously determined F(ω3) is not an unknownparameter.

The method then comprises deriving the calibration coefficients g_(a,3),u_(a,3) based on the correlations of calibrated measurements performedat 336 and of the uncalibrated measurements performed at 334 (block348). This operation is performed first by determining the complexparameters p_(a31) and q_(a31) and then by determining the coefficientsg_(a,2), u_(a,2) based on p_(a21) and q_(a21), F(ω3) and g_(a,1) andu_(a,1) using the correlations as well as the relationship for eachfrequency between calibrated and uncalibrated measurements (see equation(5) and (6) above).

Once the coefficients have been determined for each of the sensors andeach of the frequency, the measurements taken in the formation may becorrected (block 349) using the coefficients g_(a,b) and u_(a,b) thathave been determined previously during the calibration. If a sensor aobtains a measurement Zmeas at a frequency fb, the measurement will becorrected using the coefficients g_(a,b) and u_(a,b) as follows in orderto correct the measurement and then determining the resistivity of theformation based on such measurement:

Z _(corr) =g _(a,b) Z _(meas) +u _(a,b)

The method that is presented above enables to calibrate automaticallythe sensor in the formation, taking into account the parameters of theborehole environment and without any previous operation or control fromthe operators at the surface.

Alternatively, it is also possible to calibrate the sensors whenmeasurements are taken in a formation where at least two of theoperating frequencies are mainly sensitive to the mud. This is possiblefor instance when a formation has a very low resistivity. Indeed, whenthe apparent resistivity is less than a threshold the apparent impedancestill has a linear behavior, at least at the lowest frequencies. Anexemplary method 400 for using the open hole section is disclosed inrelationship with FIG. 9.

The method 400 may include, if there is no casing or calibration cannotbe performed based on the casing in view of the borehole or jobparameters, measuring the resistivity Rt with an additional sensor(block 402). Any known sensor and resistivity determination method maybe used.

The formation resistivity Rt is then compared to a predeterminedthreshold (block 406). The threshold is generally below 5 ohm·m. Thisoperation enables indeed to identify a portion of the wellbore in whichthe measurement has a linear behavior, at least for frequencies belowthe threshold as defined above.

If the formation resistivity measured with the additional sensor isbelow the predetermined threshold, the calibration method may beperformed for the whole set of measurements for which the condition ismet. In other words, the calibration set is selected (block 408) as perthe condition of operation 406. This operation then corresponds tooperation 310 of method 300. The method 400 may then comprise theoperations 312-349 of method 300.

Additional verification may be performed before launching thecalibration. For instance, the method may also comprise representing inthe complex plane the measured impedance over a depth intervalcorresponding to the interval at which resistivity is under thethreshold. If the representation of the impedance measurements is linearover the interval, at least at two frequencies, the calibration methodmay be launched.

In this case, there are however a few changes. In particular, the methodincludes expressing the calibration parameters as in operation 318 butthe expression is different compared to the one of method 300. Theexpression indeed varies from when the calibration is performed in thecasing as the model still has to take into account the formationresistivity. The calibrated measurement is expressed as follows:

ZAPP_CAL=ZAPP_MUD+kf·Rt

The calibration measurement may also be expressed as follows:

ZAPP_CAL=g·ZAPP_UNCAL+u

Therefore, in this case, the measurement corresponding to mud impedancethat enables to perform the calibration shall be expressed as:

ZAPP_MUD=g·ZAPP_UNCAL+(u−kf·Rt)

Wherein ZAPP_MUD is the calibrated measurement accounting for the mudimpedance (unknown), ZAPP_UNCAL is the uncalibrated measurement, g and uare calibration parameters, Rt is the formation resistivity (determinedat operation 404) and kf is a geometrical factor having a known value atlow resistivity (its value may for example be determined by simulationor modelling).

More generally a method 500 according to the disclosure (shown on FIG.10) includes taking (block 502) apparent impedance measurements with thesensor at a set of frequencies comprising at least one frequency and ata first plurality of locations in the borehole, wherein the measurementare uncalibrated measurements. The measurements may be taken at severalfrequencies and the method may be launched in real-time or after themeasurements have been acquired for the whole borehole. The method maythen include identifying a portion of the borehole in which the apparentimpedance measurements has a predetermined behavior at least at a firstfrequency (block 504), wherein the predetermined behavior is that theapparent impedance measurements taken in the portion are substantiallyfitting a linear model when represented in the complex plane. Theportion may be a cased hole portion or an open hole portion penetratinga formation having a resistivity below a predetermined threshold. In thelatter case, identifying the portion may include estimating a formationresistivity based on measurements taken with one or more additionalsensors and comparing the measured resistivity to the threshold.Alternatively, the method may include representing apparent impedancemeasurements taken at the at least one frequency in a complex plane, inparticular the measurements taken at the second plurality of locations,and fitting a line to the plurality of measurements obtained at a secondplurality of locations. Identifying the portion may include verifyingthat the standard deviation of the measurement points compared to thefitting line is under a threshold. When the set comprises a plurality offrequencies, identifying the portion may also include comparing theslope of a fitting line obtained for measurements taken at the firstfrequency to a fitting line obtained for measurements taken at a secondfrequency of the plurality. The measurements taken at the secondplurality of locations is a calibration set, and the method includesvalidating the calibration set, by verifying if one or more criteriarelative to the set are met. Such criteria may relate to the standoff,in order to verify that the standoffs are diverse and provide a robustcalibration.

The method then includes using a plurality of measurements obtained at asecond plurality of location situated in said portion at the firstfrequency to determine calibration coefficients for the measurementstaken at the first frequency (block 506). The second plurality oflocations may be a subset of the first plurality of locations. Thecalibration coefficients may be defined as followsZAPP_CAL_Fb=g_(b)·ZAPP_UNC_Fb+u_(b), wherein ZAPP_CAL_Fb is a calibratedmeasurement at frequency fb, wherein ZAPP_UNC_Fb is the uncalibratedmeasurement at frequency fb and wherein g_(b) and u_(b) are thecalibration coefficients for frequency fb. When the downhole toolcomprises a plurality of sensors, the calibration coefficients aredetermined separately for each sensor.

Operation 506 may include selecting a predetermined point on thecorresponding fitting line and calculating a first relationship betweenthe calibration coefficients so that this point corresponds to theorigin of the complex plane when the measurements are calibrated. It mayalso include determining a correlation between uncalibrated measurementand theoretical measurement in a non-dispersive medium (such as air) andcalculating a second relationship between the calibration coefficientsfor the first frequency based on the correlation. The calibrationcoefficients may be determined based on the first and the secondrelationship. The first frequency is therefore below a first threshold,the threshold may be 10 MHz

When the method includes a plurality of frequencies, it may also includeusing the calibration coefficients for the measurements at the firstfrequency to determine the calibration coefficients for the measurementsat a second frequency (block 506). This operation may for instanceinclude correlating uncalibrated measurements taken at the firstfrequency to uncalibrated measurements taken at the second frequencywith the following expression:

ZAPP_UNC_F2=p ₂₁·ZAPP_UNC_F1+q ₂₁

Wherein ZAPP_UNC_F2 is a uncalibrated measurement taken at the secondfrequency, Wherein ZAPP_UNC_F1 is a corresponding uncalibratedmeasurement taken at the first frequency, Wherein p₂₁ and q₂₁ areunknown coefficients, wherein the method further comprises calculatingp₂₁ and q₂₁ using the measurements taken at the second plurality oflocations at first and second frequencies.

The operation 506 may also include correlating calibrated measurementstaken at the first frequency to calibrated measurements taken at thesecond frequency with the following expression:

${{ZAPP\_ CAL}{\_ F2}} = {{ZAPP\_ CAL}{\_ F1} \times \frac{F({\omega 2})}{\omega \; 2} \times \frac{\omega \; 1}{F( {\omega \; 1} )}}$

-   -   wherein ZAPP_CAL_F2 is a uncalibrated measurement taken at the        second frequency, wherein ZAPP_CAL_F1 is a corresponding        uncalibrated measurement taken at the first frequency, wherein        ω1 and ω2 are the radial frequencies respectively corresponding        to the first and second frequencies and wherein F(ω) is a        dispersion function of a drilling fluid filling the borehole.

When the second frequency is below the predetermined threshold, theoperation 506 may include determining a correlation between uncalibratedmeasurement and theoretical measurement in a non-dispersive medium andcalculating a first relationship between the calibration coefficientsbased on the correlation, as well as determining a second and a thirdrelationship between the calibration coefficients and the value F(ω2) ofthe dispersion function at the second frequency based on the correlationbetween uncalibrated measurements at first and second frequencies andcorrelation between calibrated measurements at first and secondfrequency, wherein the method includes determining the calibrationcoefficients and value F(ω2) of the dispersion function at the secondfrequency based on the first, second and third relationships.

Alternatively, the method may include, in particular when the secondfrequency is above the predetermined threshold, modelling the dispersionfunction F(ω) according to a predetermined model and determiningparameters of the model for instance based on values of the dispersionfunction obtained at least at two reference frequencies (generally belowthe threshold). When the tool comprises several sensors at least a valueof the dispersion function used in the modelling is a combination of thevalues obtained for each of the plurality of sensors. In this casedetermining a value of the dispersion function at said referencefrequency is based on the uncalibrated measurements at said referencefrequency and includes determining a correlation between uncalibratedmeasurement and theoretical measurement in a non-dispersive medium (asexplained above). The calibration coefficients for the second frequencyare then calculated based on the correlation of the between uncalibratedmeasurements at first and second frequency and correlation betweencalibrated measurements at first and second frequency.

The method may also include (block 510) correcting the (uncalibrated)apparent impedance measurements using the calibration coefficients. Themeasurements taken at all of the plurality of locations in the boreholemay be corrected using the coefficients determined as defined above. Theresistivity of the formation is determined based on the corrected orcalibrated measurements.

The method according to the disclosure is generally performed in aborehole containing oil-based mud and is particularly appropriate for aLWD tool for which the standoff with the borehole wall is moreimportant. The method according to the disclosure provides an automatedon-site calibration that does not require intervention of the operatorbefore or during the measurement acquisition.

The disclosure also relates to an apparatus for calibrating aresistivity measurement, wherein the apparatus includes a downhole toolconfigured to be conveyed in a borehole, and having at least one sensorsituated at a non-zero standoff distance from the borehole configured toestimate the resistivity of an underground formation penetrated by theborehole by taking apparent impedance measurements at a set offrequencies comprising at least one frequency at a first plurality oflocation in the borehole, wherein the measurement are uncalibratedmeasurements. The apparatus also includes a set of processors includingone or more processors configured to identify a portion of the boreholein which the apparent impedance measurements at least at a firstfrequency of the set have a predetermined behavior, wherein thepredetermined behavior is that the apparent impedance measurements atthe at least one first frequency taken in the portion are substantiallyfitting a linear model when represented in the complex plane, and use aplurality of measurements obtained at a second plurality of locationsituated in said portion at the first frequency to determine calibrationcoefficients for the measurements taken at the first frequency.

The downhole tool may be a logging while drilling tools. It may alsocomprise several sensors. In the latter case, the set of processors isconfigured to determine calibration coefficients at the first frequencyfor each of the sensor. The sensor may be configured to take apparentimpedance measurements at a plurality of frequencies. In the lattercase, the set of processors may be configured to use the calibrationcoefficients for the measurements at the first frequency to determinethe calibration coefficients for the measurements at a second frequency.The set of processors may be situated downhole, at the surface, remotelyfrom the rig or partially downhole, and/or partially at the surfaceand/or partially remotely. They may be configured to execute one or moreoperations of the method as disclosed above.

The disclosure also relates to a computer readable storage mediumcomprising instructions to identify, based uncalibrated apparentimpedance measurements at a set of frequencies comprising at least onefrequency at a first plurality of location in a borehole by a downholetool conveyed in the borehole, a portion of the borehole in which theapparent impedance measurements at least at a first frequency of the sethave a predetermined behavior, wherein the predetermined behavior isthat the apparent impedance measurements at the at least one firstfrequency taken in the portion are substantially fitting a linear modelwhen represented in the complex plane, and using a plurality ofmeasurements obtained at a second plurality of location situated in saidportion at the first frequency to determine calibration coefficients forthe measurements taken at the first frequency.

Generally, the computer storage medium comprises instructions forperforming one or more operations of the method as mentioned above.

The systems and methods introduced in the present disclosure aresusceptible to various modifications, variations, and/or enhancementswithout departing from the scope of the present disclosure. For example,different configurations can be employed for the resistivity tool toaccommodate a downhole tool or other challenging environment.Accordingly, the present disclosure expressly encompasses all suchmodifications, variations, and enhancements within its scope.

The foregoing outlines features of several embodiments so that a personhaving ordinary skill in the art may better understand the aspects ofthe present disclosure. A person having ordinary skill in the art shouldappreciate that they may readily use the present disclosure as a basisfor designing or modifying other processes and structures for carryingout the same purposes and/or achieving the same advantages of theembodiments introduced herein. A person having ordinary skill in the artshould also realize that such equivalent constructions do not departfrom the spirit and scope of the present disclosure, and that they maymake various changes, substitutions and alterations herein withoutdeparting from the spirit and scope of the present disclosure.

The Abstract at the end of this disclosure is provided to comply with 37C.F.R. § 1.72(b) to allow the reader to quickly ascertain the nature ofthe technical disclosure. It is submitted with the understanding that itwill not be used to interpret or limit the scope or meaning of theclaims.

1. A method for calibrating a resistivity measurement taken by adownhole tool in a borehole, wherein the downhole tool estimates theresistivity of an underground formation penetrated by the borehole withat least one sensor situated at a non-zero standoff distance from theborehole, wherein the method includes: taking apparent impedancemeasurements with the at least one sensor at a set of frequenciescomprising at least one frequency and at a first plurality of locationsin the borehole, wherein the measurement are uncalibrated measurements,identifying a portion of the borehole in which the apparent impedancemeasurements at least at a first frequency of the set have apredetermined behavior, wherein the predetermined behavior is that theapparent impedance measurements taken in the portion are substantiallyfitting a linear model when represented in the complex plane, using aplurality of measurements obtained at a second plurality of locationsituated in said portion at the first frequency to determine calibrationcoefficients for the measurements at said frequency.
 2. The methodaccording to claim 1, including correcting the uncalibrated measurementsat the first plurality of measurements using the calibrationcoefficients.
 3. The method according to claim 1, wherein the portion ofthe borehole is a cased portion.
 4. The method according to claim 1,wherein the portion of the borehole is an open hole portion penetratinga formation having a resistivity below a predetermined threshold.
 5. Themethod according to claim 1, wherein the calibration coefficients aredefined as follows:ZAPP_CAL_Fb=g _(b)·ZAPP_UNC_Fb+u _(b) Wherein ZAPP_CAL_Fb is acalibrated measurement at frequency fb Wherein ZAPP_UNC_Fb is theuncalibrated measurement at frequency fb Wherein g_(b) and u_(b) are thecalibration coefficients for frequency fb
 6. The method according toclaim 1, including representing a plurality of apparent impedancemeasurements taken at the at least one frequency and at the secondplurality of locations in a complex plane and fitting a line to saidplurality of measurements.
 7. The method according to claim 6, whereinthe set includes a plurality of frequencies, and wherein identifying theportion includes comparing the slope of a fitting line obtained formeasurements taken at the first frequency to a fitting line obtained formeasurements taken at a second frequency of the plurality.
 8. The methodaccording to claim 7, wherein determining the calibration coefficientsfor the measurements at the first frequency includes selecting apredetermined point on the corresponding fitting line and calculating arelationship between the calibration coefficients so that this pointcorresponds to the origin of the complex plane when the measurements arecalibrated.
 9. The method according to claim 1, wherein determining thecalibration coefficients for the measurements at the first frequencyincludes determining a correlation between uncalibrated measurement andtheoretical measurement in a non-dispersive medium and calculating arelationship between the calibration coefficients based on thecorrelation, wherein said frequency is a frequency below a predeterminedthreshold.
 10. The method according to claim 1, wherein the set includesa plurality of frequencies, wherein the method includes using thecalibration coefficients for the measurements at the first frequency todetermine the calibration coefficients for the measurements at a secondfrequency of the set.
 11. The method according to claim 10, includingcorrelating uncalibrated measurements taken at the first frequency f1 touncalibrated measurements taken at the second frequency f2 with thefollowing expression:ZAPP_UNC_F2=p ₂₁·ZAPP_UNC_F1+q ₂₁ Wherein ZAPP_UNC_F2 is a uncalibratedmeasurement taken at the second frequency f2, Wherein ZAPP_UNC_F1 is acorresponding uncalibrated measurement taken at the first frequency f1,Wherein p₂₁ and q₂₁ are unknown coefficients, Wherein the method furthercomprises calculating p₂₁ and q₂₁ using the measurements taken at thesecond plurality of locations at the first and second frequencies. 12.The method according to claim 10, wherein it includes correlatingcalibrated measurements taken at the first frequency f1 to calibratedmeasurements taken at the second frequency f2 with the followingexpression:${{ZAPP\_ CAL}{\_ F2}} = {{ZAPP\_ CAL}{\_ F1} \times \frac{F({\omega 2})}{\omega \; 2} \times \frac{\omega \; 1}{F( {\omega \; 1} )}}$Wherein ZAPP_CAL_F2 is a calibrated measurement taken at the secondfrequency f2, Wherein ZAPP_CAL_F1 is a corresponding calibratedmeasurement taken at the first frequency f1, Wherein ω1 and ω2 are theradial frequencies respectively corresponding to the first and secondfrequencies f1 and f2 and wherein F(ω) is a dispersion function of adrilling fluid filling the borehole depending on radial frequency ω. 13.The method according to claim 12, including modelling the dispersionfunction F at radial frequency ω according to a predetermined model anddetermining unknown parameters of the modelled dispersion function basedon values F(ω1), F(ω2) of the dispersion function obtained at least atreference frequencies below a predetermined threshold.
 14. The methodaccording to claim 1, wherein the downhole tool comprises a plurality ofsensors, wherein the calibration sensors are determined separately foreach sensor.
 15. The method according to claim 14, wherein the downholetool comprises a plurality of sensors, wherein the calibrationcoefficients are determined separately for each sensor, wherein at leastan unknown parameter of the dispersion function model is a statisticalcombination of the unknown parameter values obtained for each of theplurality of sensors.
 16. The method according to claim 1, wherein themeasurements taken at the second plurality of locations is a calibrationset, wherein the method includes validating the calibration set, byverifying if one or more criteria relative to the set are met.
 17. Anapparatus for calibrating a resistivity measurement, wherein theapparatus includes: a downhole tool configured to be conveyed in aborehole, and having at least one sensor situated at a non-zero standoffdistance from the borehole and configured to estimate the resistivity ofan underground formation penetrated by the borehole by taking apparentimpedance measurements at a set of frequencies comprising at least onefrequency and at a first plurality of locations in the borehole, whereinthe measurement are uncalibrated measurements, a set of processorsincluding one or more processors configured to: identify a portion ofthe borehole in which the apparent impedance measurements at least at afirst frequency of the set have a predetermined behavior, wherein thepredetermined behavior is that the apparent impedance measurements atthe at least one first frequency taken in the portion are substantiallyfitting a linear model when represented in the complex plane, using aplurality of measurements obtained at a second plurality of locationsituated in said portion at the first frequency to determine calibrationcoefficients for the measurements taken at said frequency.
 18. Theapparatus of claim 17, wherein the downhole tool is a logging whiledrilling tool.
 19. The apparatus of claim 17, wherein the downhole toolcomprises a plurality of sensors and the set of processors is configuredto determine calibration coefficients at the first frequency for each ofthe sensor.
 20. The apparatus of claim 17, wherein the sensor isconfigured to take apparent impedance measurement at a plurality offrequencies and the set of processors is configured to use thecalibration coefficients for the measurements at the first frequency todetermine the calibration coefficients for the measurements at a secondfrequency.